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Throughout his analysis, he assumed that the molecular size Maxwell’s Improved Kinetic Theory,
was sufficiently small so that s << L and s << N−1/3, the mean 1867 molecularspacing.HeproceededtocalculatAetrhtiecltera2nsportEMquaaxtwioenlls(1867)reformulatedkinetictheorybyconsidering
properties having direct relevance to acoustical phenom- ena. He found the shear viscosity (μ) to be μ = ρL<v>/3 = (M<v>/s2)/(3π√2), which does not directly depend on the P of the gas. (He noted that lack of dependence on P with some surprise, first in a letter to G. G. Stokes in May 1859.) This required considering collisions between molecules
rticle 2from aEdjqauceantitognass regions having different flow velocities.
He also obtained predictions for thermal conductivity and,
for mixtures, gas mass diffusivity. (By 1862, Rudolf Clausius
published a significant criticism of the result for thermal
conductivity, quickly appreciated by Maxwell as a proper
the consequences of binary molecular encounters, provid- ing a rigorous background for subsequent advance. He gave a new derivation of the velocity distribution function now denoted by f(v), giving a result equivalent to Equation 1.
concern.) More relevant to the physics of sound, however,
the collision. He also used the conservation of translational kinetic energy in that derivation. Of greater importance, however𝛾𝛾, w=as𝑐𝑐h!i/s𝑐𝑐n!ew=ap1p+ro2ac/(h3tβo)calculating transport co- (2) efficients through the introduction of the following transfer equation for the rate of change in a quantity of interest
α
constant pressure and volume (c and c , respectively) for (δQ/δt)dN = (Q − Q)Vb db dφ dN dN
−3 −1/2 2 2 2
was Maxwell’s dprNed=ic4tiNon foπr thev reatxiop(o−fvsp/αeci)fidcvheats at a
the gas
𝛾𝛾 = 𝑐𝑐 /𝑐𝑐 = 1 + 2/(3β) (2) !!
(3) (3)
where the quantities of interest (Q) are functions of the
Cartesian(2co)mponents of velocity referred to a coordinate
system connecting the molecular centers of force at the dis-
tance of closest approach, φ is the azimuthal angle, b is the
impact offset radius associated with the collision, and V is
the relative velocity. From the laws of mechanics, Maxwell
where for the spherical gas particles in thermal equilibrium
(β − 1) = (rotational energy)/(translational energy). For
rough spheres, from Maxwell’s equipartition of the energy
(3)
principle, he found β = 2 and γ = 4/3. He noted, however, that for air, γ had been measured to be approximately 1.408, a result that he took to be “decisive against the unqualified acceptation of the hypothesis that gases are such a system of hard elastic particles.” By 1860, Maxwell was certainly aware that Laplace’s assumption of negligible heat flow dur- ing sound propagation in gases results in the speed of sound at audible frequencies of c = √(γP/ρ). He included that re- lationship a decade later in his general textbook Theory of Heat (Maxwell, 1871).
expressed the scattering angle after each collision in terms of an integration involving b, V, the molecular masses, and the repulsive force law of interaction that he took to be in the form K0/rn, where K0 is a constant and n is an integer. Evalu- ation of the desired rate (δQ/δt) required integration over φ and b such that he was able to obtain analytical results for the desired transport coefficients: shear viscosity (μ), ther- mal conductivity (K), and mass diffusivity (D) by restrict- ing attention to n = 5. (This restriction was needed to bring about a cancellation of the integrand’s dependence on V.) Of the results obtained for these special “Maxwell molecules,” in acoustics the following relationship between K and μ is especially significant: K = Fμcv, where cv is the specific heat at constant volume and F is the dimensionless Maxwell-Euck- en factor (using modern terminology). For gases having γ = 5/3, Maxwell’s method correctly gives F = 5/2. (Because of an algebraic error first reported by Ludwig Boltzmann in 1872, this differs from Maxwell’s claimed result in his Equation 149 where taking γ = 5/3 gives F = 5/3.) Acousticians today prefer to use the Prandtl number (Pr) in the modeling of thermoacoustic devices: Pr = γ/F, which becomes 2/3 when γ = 5/3, in close agreement with modern measurements of the Prandtl number for monatomic gases.
Maxwell’s Confirmation of the Pressure Independence of Viscosity, 1866
To investigate the predicted independence of μ on pressure, Maxwell developed a new method of measuring viscosities by measuring the damping rate of rotational oscillations of layers of disks near stationary plates separated by gas. These experiments on the friction of gases were done in his resi- dence in London with assistance from his wife. The mea- surements confirmed the predicted independence on pres- sure (Maxwell, 1866).
The condition f(v1) f(v2) = f(vf1) f(vf2) produced equilibrium,
−3 −1/2 2 2 2
where dvNan=d4vNcorπrespovndextpo(t−hve /iαnit)iadlvand final velocity (1)
(1) Pv1f21
(δQ/δt)dN = (Q − Q)Vb db dφ dN dN 1f21
α
f
vectors of molecules 1 and 2, respectively, associated with
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